Moreover, we note that if the time-inverted \(\ell \) -th Newman–Penrose charge happens to vanish as well, we can repeat the process, and prove asymptotics that decay one power faster. Using our results, one can also try to recover expressions similar to the ones given for \(I_0^{(1)}\)  in [1]}, for compactly supproted data. In this case, for time-symmetric data (i.e. \(T\psi |_{\Sigma _0} = 0\) ) that also vanish at the event horizon, the time-inverted \(\ell \) -th Newman–Penrose charge will vanish as well, hence in that situation we can show the polynomial late-time law of \(\tau ^{-2\ell -4}\)  (this should be contrasted with the suboptimal rates obtained for such data in [2]} and [3]}).
